Standards
NCTM’s Principles
and Standards for School Mathematics (2000)
Geometry [Grades 6-8 (p. 233), Grades
9-12 (p. 309)]
“Specify locations and describe spatial relationships
using coordinate geometry and other representational systems.”
“Apply transformations and use symmetry to
analyze mathematical situations.”
“Use visualization,
spatial reasoning, and geometric modeling to solve problems.”
Communication [Grades 6-8 (p. 268), Grades 9-12 (p. 348)]
“Use
the language of mathematics to express mathematical ideas
precisely."
Conference
Board of the Mathematical Sciences' The Mathematical Education of Teachers
Recommendation 1. “Prospective
teachers need mathematics courses that
develop a deep understanding of the mathematics they will teach.” (Chapter
2)
Recommendation
3. "Courses on fundamental ideas of school mathematics
should focus on a thorough development of basic mathematical ideas.
All courses designed for prospective teachers should develop careful
reasoning and mathematical "common sense" in analyzing conceptual
relationships and in solving problems." (Chapter 2)
Recommendation
4. "Along with building mathematical
knowledge, mathematics courses for prospective teachers should develop the
habits of mind of a mathematical thinker and demonstrate
flexible, interactive styles of
teaching." (Chapter 2)
Materials and Technologies
Activity Sheets
Graph paper
Graphing Calculator
Historical
Background
Arthur Cayley (1821-1895) and James
Sylvester (1814-1897) were two of the earliest mathematicians to writer about
transformation theory. In 1872, Felix
Klein classified geometries by applying this definition: A geometry is the study of invariant properties
of a set of points under a group of transformations. Thus, Euclidean geometry is the study of invariant properties, such
as angle measure, area, and parallelism of sets of points under the group
of motions. Josiah Willard Gibbs (1830-1903),
an American physicist, developed vector analysis in 1881 where vectors have
both magnitude and direction associated with them. One geometric application of various motions
of the plane is in simple geometric designs that form plane-filling repeated
patterns. The Dutch artist M. C. Escher
extended this idea to very intricate plane-filling patterns involving such
figures as fish, birds, horsemen, and reptiles.
One of the most recent and important
applications of transformations has been in the new of computer graphics. Computer graphics refers to the use of the
computer to store, manipulate, and show pictorial information. For a typical computer graphics program, the
output will be a picture composed of a set of points. A matrix is a rectangular arrangement of objects. They can be used to describe various geometric
figures and their transformations. These
matrices are also used to describe the pictures seen on the television and
computer monitors as rectangular arrays of square dots.
Instructor’s Notes
- Expected Time: Activity #1: Sets of Equations for Transformations should take about 30 minutes.
(Exercises at end could be assigned as homework.) Activity #2: Transformations with Matrices should take about 45 - 60 minutes. (Exercises at end could be assigned as homework.)
§
Materials:
Graph paper
Graphing calculator
Activity
Sheets
Special
Matrices list as a handout at completion of exercise
- Grouping: Activities
can be done in groups or individually if desired.
§
Mathematics:
Activity #1: In this activity
students are asked to explore translations and reflections about the axes
using sets of linear equations. Examples of this are provided below. The graphing
calculator is suggested as a way to generate points of the pre-image points
in lists. Using the linear equations, students can then generate the image
points in a second set of lists.
a)
If P(3,4) has the image P’(7,9), then the translation may be
represented by the equations. 
b)
Here is another example using the set of equations above.
c)
Another example: Plot and connect points A(2,1), B(5,1), and
C(5,7). Translate this triangle algebraically
with a vector that is right 5 and up 3. In other words, 
. Sometimes
we denote this vector similar to an ordered pair but using brackets, [5,3].
d)
Another example: Find
the image of 2x +4y – 6 = 0 under the translation with equations 
and 
.
Solving for x
and y yields 
Thus 2x + 4y
– 6 = 0 has the image 2(x’ – 7) + 4(y’ – 2) – 6 = 0 and simplifies to 2x’ + 4y’ – 28 = 0
e)
For reflection across the x-axis the equations are 
and for reflection across the y-axis the equations are 
An example on rotations is included on the student
activity sheet.
Activity #2: The
field of computer graphics is one of the most recent and important applications
of transformations. A figure is transferred from coordinates in the real world
to screen coordinates in computer graphics. These screen coordinates must
fit in the viewing window and are often referred to as pixels. Computer graphics refers to the use of the
computer to store, manipulate, and show pictorial information. With computer software the output will be a
picture composed of a set of points. In
computer graphics, a point in two dimensions can be represented as a column
matrix 
. For example the point (1,5) is represented as the matrix 
.
A brief review of matrices and matrix multiplication is necessary here.
Instructor should also make sure that students can perform matrix operations
on the graphing calculator. Many transformations of points in two dimensions
can be represented as 2´2 matrices, with two rows and two columns. The students are asked to find the matrix that will perform the
requested transformation. A list of Special 2x2 Matrices
is included but should not be given to the students
until they have completed the activity.
Closure
a)
Discuss with students the various transformations and the technologies
that were explored in these activities.
b)
How did each technology enhance their understanding of the
transformations and isometries?
c)
How could they use their experiences with these activities
in their future classroom?
d)
Did these activities strengthen their understanding of transformational
geometry?
§
Connections/ Discussions/
Further Explorations
a) Have students define each of the three isometries in
their own words
b) Research the use of transformational geometry in today’s
technological world and some real world applications of matrices and transformations
c) Explore transformations using only a compass and straight
edge, thus reinforcing the critical ideas of each transformation
d) Explore similarity transformations
e) Find the curriculum guidelines for your state or local
school district and look at the course objectives for a particular grade level
or high school course. How many objectives
were met in “Transformations Everywhere?”
f)
Have students describe the
various technologies used and discuss how these technologies affected their
understanding of transformations.
Links
http://www.netcomuk.co.uk/~jenolive/homegrps.html
http://forum.swarthmore.edu/dynamic/jrk/ferris_dir/
http://www.hannasd.k12.pa.us/sths/departments/mathdept/Yohe2/trans.htm
http://www.woodrow.org/teachers/mi/1993/23bann.html
http://www.uh.edu/~hollyer/Module8/m8ppt/sld003.htm
References
1.
Smart, James. Modern
Geometries, Brooks/Cole Publishing Co,
2.
"Advanced Algebra" --Holt, Rinehart and Winston,
Inc.
3.
"Contemporary Mathematics in Context -- A Unified Approach
(Course 2 Part. A)" Everyday Learning Corp.
4.
Pettogrezzo, A. Matrices and Transformations, 1966,
Dover Publications, Inc.: NY.
5.
UCSMP, Advanced Algebra
6.
Edwards, Michael Todd. “Visualizing
Transformations: Matrices, Handheld Graphing Calculators, and Computer Algebra
Systems.” The Mathematics Teacher, Vol. 96, No. 1, Jan, 2003.
