1) Tracing Paper. Miras.
Date: Fri, 24 Jan 1997 20:02:53 -0500 (EST) From: Bheinrich@aol.com To: uc@mathforum.org Subject: Re: teaching transformations I introduce the year with an informal unit on transformations. I have found that this approach helps the students visualize better all year. I hand out squares of tracing paper to each student. They have no trouble using this approach, and I am able to use the language of transformations when teaching about triangles and other polygons. When we first started working with congruent triangles, the students listed the type of transformation used to make the two triangles coincide in each situation. I keep Miras on hand for reflections and also used the overhead TI92 to demonstrate various rotations and reflections. Barbara Heinrich Wauconda High School, Wauconda, IL.2) Geometer's Sketchpad
From: Dmiller155@aol.com Date: Sat, 25 Jan 1997 17:25:09 -0500 (EST) To: uc@mathforum.org Subject: Re: teaching transformations The program "Geometer's Sketchpad" can be used very nicely to help students visualize transformations. The literature that comes with the program suggests ways to have students explore transformations.3) Patty Paper Geometry
Date: Sat, 25 Jan 1997 22:03:07 -0500 From: "Guy F. Brandenburg" guyfbran@erols.com Organization: DCPS&NCTM To: Marivel Miranda (uc@mathforum.org) Subject: Re: teaching transformations I've had trouble in the past as well, but recently I ordered a book called Patty Paper Geometry by Michael Serra (who wrote Discovering Geometry, published by Key Curriculum Press). It had a number of good ideas, including some on rotations. I don't do it exactly the way he describes, but the general idea is that with sheets of relatively cheap semi-transparent hamburger-patty-paper, it is easy to do rotations, reflections, and translations. Using this stuff, it is even fairly easy to show that the result of two reflections across two different, intersecting lines is a rotation that has as its center the point where the two lines meet, and that the angle of this rotation is twice the angle from the first line to the second line. You can buy patty paper from various math ed supply houses in boxes of 1000 sheets for about 6 bucks. I suppose it would be cheaper from a restaurant supply house, but I don't know any of them. For the simple rotation, what I had the students do was this: 1. Using a pencil (works much better than a pen) draw a simple picture somewhere near one edge of one sheet of patty paper--something NOT symmetrical. (This way everybody draws something different: cats, umbrellas, houses, dogs, boats, fancy letters, etc.) 2. Draw a point C somewhere else on the paper. 3. Draw a ray using C as its endpoint. 4. Draw another ray, again using C as the endpoint, forming an angle. 5. Now put another piece of patty paper on top of the first. Trace your picture, the center point C, and the first ray. (don't bother with tracing the second ray) 6. Now, keeping both papers lined up, put the tip or the eraser onto point C, to keep the center of rotation fixed. Rotate the top sheet until the first ray lines up with the center ray. You have now performed the desired rotation. 7. While they are doing this at their desks, I am modeling this on the overhead projector with two plastic transparencies and with erasable overhead markers. 8. To make a permanent example of how the first image got rotated, then put the top paper on the bottom, line things up properly again (pre- and post-transformation), and then trace where the original object (the pre-image) has been rotated to. 9. Then we do the same thing for a couple of random points, draw the line segments connecting them, and so on. Then we prove stuff about what happens on the blackboard. 10. You can also use patty paper to rotate things in increments of 90 degrees on the cartesian rectangular coordinate plane. 11. I still have yet to give a unit test to see if it produces good results, since I did this quite recently, but it seems to me, unscientifically, that the students got a much better feel for what they were doing than the way I was doing before. Before, a lot of students would openly complain that they just didn't understand. This time, that's not happening at all, and they really enjoy drawing the little pictures. 12. I'm not sure exactly how I will test it. Some I will do on x-y axes and ask where points go after 90 degree CW or CCW rotation around the origin, or after 180 degree rotation. I guess I will also give them patty paper and some little drawing of my own, photocopied or dittoed onto the test paper, and ask them to rotate the drawing a specified number of degrees. Do you want progress results?