Translations, Reflections, Rotations Oh My!!
Math Curriculum Day August 21, 2009 Genny Simpson Early College High School
1.
Matrix multiplication At the Early College, we teach algebra II before geometry. This order makes teaching transformations of geometric figures less time consuming and less frustrating, as if that’s even remotely possible! In addition to reviewing multiplication, I also remind them about the identity matrix. I give my students a graph of an image and its pre-image. We put the coordinates of their vertices in point matrices. We write a matrix multiplication equation in order to determine the matrix we would use to produce the given image from its pre-image. The students themselves determine the entries in the transformation matrix. I prefer this “discovery” method to giving the students the matrices they need to perform each transformation. After find the transformation matrix, they write the rule themselves. Being able to reason and think backwards proves beneficial when students can’t remember which matrix produces which transformation. B' A
A' B
A
B
2 ⋅ 3
4
C
A'
1 −3 − 1 = 1 − 1 2 4
C'
C
B'
90° counterclockwise rotation
3.02 Use matrix operations (addition, subtraction, multiplication, scalar multiplication) to describe the transformation of polygons in the coordinate plane.
I repeat this process with every reflection and rotation where matrix multiplication is required.
C'
1 1
2.
Perpendicular Line and Patty Paper Method I am always looking for different ways to teach transformations. I thought of this idea to show 90 degree rotations clockwise or counterclockwise about the origin. I choose a vertex closest to the origin and draw the line that contains it and the origin. I then draw the line perpendicular to that line through the origin. I mark the image of the vertex I used on the perpendicular line. I trace the pre-image onto a sheet of patty paper. I label the vertices of the traced pre-image as A’, B’, and so forth. Holding the patty paper in place at the point of rotation, in this case the origin, I turn the patty paper in the direction of rotation until the vertices align. I then draw the image on the graph under the tracing. This method works with any point as the center of rotation, as in the sketch below.
6
4
A E 2
C
B
C' -10
-5
5
C'
-2
A' -4
B' -6
-8
10
My students liked this method. I asked the testing coordinator if we could use patty paper on the EOC, but I was told that we could not. As a result, I told my students to just tear a copy of the figure from their scrap paper to help them to see the rotation.
3.
Rectangles and Patty Paper Method Two of my students last spring, Amy and Leyna, devised another method for making rotations easier to see. I had not thought of this, but it is a great visual. The girls suggested shading in the rectangular region created by the horizontal axis, the vertical axis, the point of rotation, and the vertex closest to the point of rotation. If the rectangle was a 2 x 3, they rotated it, or “laid it down� as they called it, in the direction of the rotation. Rotating the rectangle moved the vertex to its new location. They finished with the patty paper, as in the previous method.
Rotate ABC 90° counterclockwise.
B'
A
C'
A'
B
Chelsea & Leyna's Method C
I hope this helps those of you teaching geometry or tech math. I know this can be quite challenging. I am also a huge fan of graphing software such as Geometer Sketchpad. All the graphs in this presentation were done using Sketchpad. It would be great if we all had access to computers so that our students could perform these transformations for themselves with the graphing software.
Please feel free to share with me any ideas you have or strategies and activities you use in your classroom. My e-mail address is genny.simpson@ucps.k12.nc.us. We are all in this together, and it is our students who benefit from our combined efforts!
M.C. Escher. 1898-1972. Artist, and leading exponent of the art of tessellation. "The geometry of space translates to a reoccurring theme in my creations: the tessellation. A tessellation is an arrangement of closed shapes that completely covers the plane without overlapping and without leaving gaps. The regular division of the plane had been considered solely in theory prior to me, some say. I diverged from traditional approaches, and chose instead to find solutions visually. Where other mathematicians used notebooks, I preferred to use a canvas. To gain access to a greater number of designs, I used transformational geometry techniques including reflections, glide reflections, translations, and rotations. The result is a ´mathematical tessellation of artistic proportions.'"
Escher Tessellation