This afternoon, we learned more about Sketchpad with Annie. A basic outline of what we did looks like this:
I'm a bit wary about outlining the entire process since it is really educational and a lot of fun, so I strongly suggest figuring it out yourself. I will say something to get you started.
Start with a horizontal line segment AB and a point C not on the segment. Connect C to a point D on AB. Draw the perpendicular bisector of segment CD. Trace this bisector as you move point D along AB. What do you see?
Now, the key is to figure out exactly what point on the bisector actually traces out the locus of the curve (what curve? the answer appears twice in the first few lines of what I have written on this page). When you find this point, you can (in Sketchpad 3.0) construct the locus of this point as D moves along AB.
The amazing thing about this new feature in Sketchpad 3.0 is that the locus that you've constructed is a permanent piece of your sketch and can be manipulated like anything else. (Whereas you'll notice that just tracing something and then animating gives you pretty pictures that disappear as soon as you do something else.)
If you have Sketchpad 3.0 and your browser is configured to open sketches (if it's not, here are directions for configuring Web browsers for the Mac), here's a sample sketch of a geometric parabola to explore.
Place points A, B and x on the x-axis (make sure they're defined as such and can move freely along the axis). Find their coordinates, and then use the calculator to display only their x-coordinates (y=0 of course).
Then, create the expression Ax^2 + B using the calculator. That is essentially our y. Selecting x and then "y" and plotting as (x,y) yields a parabola.
Doing the last step with y and then x gives the inverse. You now have a representation of parabolas on which you can visually observe the effects of changing A, B, and x!, and you can also see the line of symmetry between the graph and its inverse (try drawing it).
Here's a sample sketch of an algebraic parabola for you to play with.
Saving a sketch as something.gsp allows you to reference it like any other link on a Web page after you've put it on a server. (If you use a program like Fetch for the Macintosh you must ftp your sketch in the form of RAW DATA.)
I hope you have a chance to play with this and that you like it as much as we did.
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