Steve+Phelps

toc =Week 1=

Folding Circles
My folding circles background (in response to Linda's Week 1 screencast) can be found HERE. I found the Wholemovement book years ago and have used it in my Geometry classes.

My Preferences for Interactivities
//Think about if and how such interactives are useful and how one could use it in class or club and for what purpose (do we like small or large interactivities, specific or general, use with video projector or in laboratory or home, ...)//

I like small ones that kids can work with in less than 5 minutes. If they are short, they should probably be specific. I would like to include more formal interactivities in my class wiki.

My Triangle Solution
For my screencasts, I have become quite fond of Screencast-O-Matic. I have my students regularly create screencasts, and this is a perfect tool: Free, easily and quickly uploads, and easy to use. This is the first time I have used the webcam picture in a screencast (a new feature they are testing). I figured you should at least SEE you is on the other end of the screencast :)

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My Applet
I did not know you could make an applet (yet) with 4.0 Beta. Good to know. Some of my students are playing around with this version and will be glad to know this! If you double click on the applet, it //should// open in a separate GeoGebra window.

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After placing the applet, I noticed that the grid did not "keep" it original attributes. I also left the "axis/grid" buttons showing, and they appeared in the applet.

How I Captured a Point in a Circle
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=Week 2=

Constructing a Square Given the Diagonal
I really enjoy these types of constructions. I give the same things to my geometry students. Construct a square around a diagonal, construct a rhombus around a diagonal, or even construct a triangle around a median. I hope that these constructions would get my students to focus on the //properties// of a figure, and to interpret these properties as //tools//.

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Here is an applet illustrating the construction. I did not use GeoGebra 4.0 because the Navigation Bar would not display.

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=Week 3=

Here is the applet illustrating my construction. I will add my proof later. What is nice about this construction is that it can be used to fold a square into thirds.

In a nutshell, this construction involves constructing constructing the centroids of two triangles. These centroids will divide the diagonal into thirds.

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=What I have been up to the last three weeks!=

Well, I have spent the last three, maybe four weeks, preparing for my presentations at the T 3 International Conference last week. I did get to meet and work with Sheryl while there, so THAT was very cool (small world, eh?)


 * Anyway, one of my presentations was titled //What-If-Not Conic Constructions//. The What-If-Not problem posing strategy is based upon the book by Brown and Walter (2005) . Put another way, what happens when you make small changes to the paper-folding parabola construction (as illustrated in the applet below). I think I will try to put as much of my presentation on this wiki as possible. || [[image:whatifnotbook.jpg]] ||

The Parabola Construction
Begin with a point F and a line BC (the focus and directrix, respectively). Place point A on line BC, and construct a perpendicular to BC at A. Next, construct the perpendicular bisector of points A and F. The intersection point P of the perpendicular bisector and the perpendicular constructed from A is on the parabola. Indeed, the distance from P to A is equal to the distance from P to F.

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Our First What-If-Not Game
WHAT IF the perpendicular constructed at A was NOT perpendicular to the directrix, but instead, formed, say, a 60 degree angle with the directrix? Still using the intersection point P of this line and the perpendicular bisector of A and F, what kind of curve is this? The applet below shows this construction.

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The applet below answers this question. Drop a perpendicular from P to E. This length is obviously shorter that the length of AP, which is equal to the length of FP. In fact, the length of PE is (AP)(sin( angle )). Noticing the ratio FP:PE is always greater than or equal to 1 and constant for any given angle, the curve is a Hyperbola.

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QUESTION: Is it possible to create an ellipse using this construction??

Our Second What-If-Not Game.
WHAT IF the directrix were NOT a line, but instead, was a circle? Let's proceed with the construction in the same manner.
 * 1) Start with a circle with center O and an point F inside the circle.
 * 2) Place a point A on the circle.
 * 3) Construct a line perpendicular to the circle at point A. This is just a line through the center of the circle.
 * 4) Construct the perpendicular bisector of point A and point F.
 * 5) Construct the intersection point P of these two lines.
 * 6) Point P traces an ellipse.

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Why does this work? A proof by questions...
 * 1) Do you agree that FP = AP?
 * 2) Do you agree that the sum AP + PO never changes?
 * 3) Do you agree that this sum is just the radius of the circle?
 * 4) Taking (1) and (2), do you agree that the sum FP + PO never changes?
 * 5) Taking (3) and (4), do you agree that the sum FP + PO = r, the radius of the circle?
 * 6) Point P must lie on an ellipse with points F and O as foci!

The Third What-If-Not Game
WHAT IF, in the ellipse construction above, point F was NOT inside the circle, but was outside the circle?

Our Fourth What-If-Not Game
We will take our second game and apply it to the ellipse construction. WHAT IF the line constructed at point A was NOT perpendicular, but instead made, say, a 60 degree angle with the circle?

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