UNIT VIII
TRANSFORMATIONS: COMPASS AND STRAIGHTEDGE


Incenter:
You will be able to see how to construct a triangle inscribed to a circle by drawing angle bisectors and using the intersection as incenter to draw the inscribed circle. The approach is using the actual steps of a construction and with an animation of a completed construction.


Circumcenter:
Students are presented with the steps to construct a circumscribed circle about a circle by constructing perpendicular bisectors and using the intersection as circumcenter to draw the circumscribed circle. You will find it very didactic since it presents the actual steps with compass and ruler, and one animation once a construction was completed.

 Translations:
Students will be able to perform translations along a translation vector with compass and ruler. The movies that show animated characters use this technique in the coordinate plane and with the help of a computer.

 Translations in the Coordinate Plane:
Students will be able to perform translations along a translation vector with compass and ruler and in the coordinate plane applying a translation rule for a translation vector with translation components in x and y. In the previous lesson, you were exposed to a lesson to do the same with compass and ruler. Now, you will be able to put it into practice the same way that cartoons make it with high end computers.

 Reflections:
Reflections are performed with compass and ruler along a line of reflection. Symmetry (line of reflection) is a common feature in nature, arquitechture, and the human body itself. So after you had completed the lesson you will have a better appreciation of this feature in graphics and objects.


Reflections in the Coordinate Plane:
Reflections are performed with compass and ruler along a line of reflection and in the coordinate plane applying an algebraic reflection rule along the xaxis or the yaxis. Once, you mastered the reflections with compass and ruler the lesson as described before will allow you to perform them the same way that you may do it with the help of a computer.


Rotations: If a character brandishes a sword to his opponent in a way that he does it following a circular path in a rotation. Well this lesson will teach you the trick. Using a protractor and a ruler students will be able to perform rotations around a center of rotation using the compass to draw the arcs, the ruler to draw radii from the center of rotation to the vertices of the polygon to rotate, and the protractor to determine the radii intersecting the arcs in a given angle of rotation.


Rotations Coordinate Plane: This lesson presents how to perform rotations in the coordinate plane around the origin in angles of 90 and 180 degrees. Modern computers have an enormous ability to perform millions of calculations per second. Using this same concepts in this lesson, movie makers are able to simulate on the big screen a turbine rotating around his axle in the center of the turbine, or the wheel of a bike rotating around the axle that holds the tyre to the bike body.


APPLICATION PROBLEM: A PATH FLIGHT Transformations imply rotations, translations and reflections. All these take place when any plane crosses the sky. In this lesson you will see how they are used in aviation. Particularly we draw from some historic remembrances of ww2 of the epic DOG fights in the skies of England when the Germans bombarded the British Cities with the Blitz attacks, and they defended with the advent of the radar that allowed them to counter attack on demand. This implies only the use of the miniaturized planes used on those fights.


APPLICATION PROBLEM: GENERATING FRAMES FOR A SHORT MOVIE USING MANIPULATIVES AND TI83 PLUS (still on the market) If you have asked yourself how is it that the video games you enjoy so much are done, you may get a glimpse of the process by doing this lesson, which you may follow with a ti83 plus graphing calculator and a set of rectangular, triangular, and circular small tiles on top of a graph paper or a grid on portable whiteboard. The lesson is short, but if you get a camera and take pictures of the different frames you generate, and put them in a PowerPoint and run the slide show high speed you may see your movie in action.


USING GEOLEGS to perform congruent transformations in the coordinate plane. Hands on activity. In this activity students will join four geoboards to make a coordinate plane. A worksheet will be provided with problems requiring to perform translations, reflections and rotations in the coordinated plane. Students will have to apply the corresponding transformation rule, then using rubber bands they will make the preimage and image of the polygon to be performs the transformation on (Triangles. May be extended to any as space and practicality allow it with the geoboard set.


Basic Constructions: Using a protractor and a ruler students will complete constructions for angle bisector, perpendicular bisector, 60 degree angle, 30 degree angle, 45 degree angles, equilateral triangle, perpendicular to a point not on the line, hexagon, and more. In current times, you don't see many people working with compass and ruler when doing technical drawings. Now they do it with what are called CAD programs (Computer Assisted Design = CAD) that have all this constructions as items in the different pulldown menus of the program. Nevertheless, it is not possible to make good use of those CAD programs unless you know how to make the construction with compass and ruler. Computers are dumb. If you enter garbage, they output garbage in big quantities by the way. They need your expertise.
