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A TSBVI Outreach Tutorial.
APH Graphic Aid for Mathematics
a.k.a. Graph Board.
Part 4: Graphing an inequality using the boundary line: y=(1/2)x+3
Presented by Susan Osterhaus,
a Texas School for the Blind and Visually Impaired Outreach Math Consultant.
Now, the next thing people ask about is, "Well, how do I graph an inequality?"
Now, as far as graphing an inequality, we start again with a boundary line,
and we're gonna graph y is less than (1/2)x+3.
So, if this is y=(1/2)x+3,
and notice I'm not gonna graph this completely over.
I'm being lazy. Okay. And I let my students do this, as well.
And now what we're saying is we're not considering the points on this line.
We only want to know about the points that are actually lying under this line.
So, this is... This may be difficult just because there are so many points.
But what I tell them to do is take off the rubber,
and now this kind of looks like a dashed or dotted line,
which means none of these points are really included.
And at this point, if I'm showing y is less than (1/2)x+3,
I'm talking about all of these points down here and, infinitely, many points going off the frame.
But for right now, we're talking about shading all of these in.
And if I would've left them, they could color all these in
and say they had just graphed y is less than (1/2)x+3.
But I'm not going to let them color all over my graph board.
So, instead I just tell them to go ahead and put their hand
on that portion of the graph that would be shaded
as opposed to this portion up here.
So, there's only...
We've divided the graph board into two parts, not two equal parts.
We've just graphed...
We've just divided it into two parts: the upper part and the lower part
In this case, y is less than (1/2)x+3" would be down here.
So, all they have to do is like this, show me the graph board and I'd say, "Great."
Now, we took the rubber band off
because none of these points were supposed to be on the graph.
Sometimes we actually want to include all of these points, which are called...
This is called the boundary line.
So if we want to include all those points,
we put that back on and we say that we...
If the student would put their hand this way,
we could say that they have graphed y is less than or equal to (1/2)x+3.
You noticed I added the equal part.
As soon as you add the equal part, it includes all of these points.
I'm gonna change the equation slightly, or the inequality, excuse me,
and say y is greater than or equal to (1/2)x+3.
In that case the students would put their arm on this side
and indicate that it's all of these points and what's above
is what's in the solution set.
And again, all we do is take the rubber band on or off,
tells us whether those points are included or not.
Having the equal in there, along with the less than or greater than,
makes all of these points included.
Now, let's suppose that you have somebody who doesn't want to put their arm there,
or you want to take that digital picture that we were talking about.
In that case you would need to do something like this.
Grab whatever is at hand, and if you wanted the shading to be up here,
you could just put that piece of paper there or whatever else you might have.
Or if it was y is less than or equal to (1/2)x+3,
you could put the paper there and say, "That's my shading."
Now, some students just love pushpins so much that they might want to do it this way.
I've seen some students do it this way.
Let's say that they wanted to shade above, then they would just put lots of pushpins.
Notice how many pushpins this takes, and to me this slows the procedure down.
But some students just really loved this so much,
they just think it's so fascinating that they want to spend all this time
and make the best digital picture that was ever made.
And I'm talking about totally blind students.
Along with... Along with low-vision students.
So, they might want to do, just put all kinds of pushpins up here
and tell me that that's the shading.
Again, just putting the hand there is fast and quick for me,
but if they want that digital picture, they might want to do it that way or again.
As far as I'm concerned, and most math teachers would say the same thing,
that little piece of paper or whatever you want to show
that that's where the shading is, it's just fine.
But if you have students who just think this is more beautiful, you know,
hey, it's okay for math to be beautiful.
So, again, what we've just shown you is the graph of,
in this particular case, y is greater than or equal to (1/2)x+3.
And if we took the rubber bands off,
we would now have y is greater than (1/2)x+3.
And boy, by this time, it's starting to...
It's starting to be hard to tell where whatever is.
Except that looks pretty straight still to me.
So it does not include those particular boundary points
but everything above.
**Captions by Project readOn**