2.12 Transformations
Translations
Translation or Transformation –The words translation and transformation look and sound quite similar to students
at first. Emphasis their relationship. A translation is just one of the many transformations the students will be learning
about in this chapter.
Point or Vector –There are two mathematical objects being use in this lesson that have extremely similar looking
notation. An ordered pair is use to represent a location on the coordinate plane, and it also is used to represent
movement in the form of a vector. Some texts represent the translation vector as a mapping. The vectorv= (− 3 , 7 )
would be written(x,y)→(x− 3 ,y+ 7 ). This makes distinguishing between the two easier, but does not introduce
the student to the important concept of a vector. It can be used though if the students are having a really hard time
with notation.
The Power of Good Notation –There is a lot going on in these exercises. There are the points that make the
preimage, the corresponding points of the image, and the vector that describes the translation. Good notation is the
key to keeping all of this straight. The points of the image should be labeled with capital letters, and the prime marks
should be used on the points of the image. In this way it is easy to see where each point has gone. This will be
even more important when working with more complex transformations in later sections. Start good habits now. A
vector should be named with a bold, lower case letter, usually from the end of the alphabet. Just writing( 9 , 6 )is
a bit ambiguous, but labeling the vectoru= ( 9 , 6 ), will make the meaning clear. In time students will be able to
understand the meaning from the context, but when they are first learning good notation can avoid confusion and
frustration.
Use Graph Paper and a Ruler –When making graphs of these translations by had, insist that the students use
graph paper and a ruler. If students try to graph on binder paper, the result is frequently messy and inaccurate. It is
beneficial for students to see that the preimage and image are congruent to reinforce the knowledge that a translation
is an isometry. It is also important that students take pride in producing quality work. They will learn so much more
when they take the time to do an assignment well, instead of just rushing through the work.
Translations of Sketchpad –Geometers’ Sketchpad uses vectors to translate figures. The program will display the
preimage, vector, and image at the same time. Students can type in the vector and can also drag points on the screen
to see how the image moves when the vector is changed. It is a quick and engaging way to explore the relationships.
If the students have access to Sketchpad and there is a little class time available, it is a worthwhile activity.
Matrices
Rows Then Columns –The dimensions of a matrix are stated by first stating the number of rows and then the number
of columns. It may take some time for the students to remember this convention. Give them many opportunities
to practice. In this lesson it is important to state the dimensions correctly because only matrices with the same
dimensions can be added. The next lesson requires students to decide if two matrices can be multiplied. The order
of the dimensions is critical in making that determination.
Additional Exercises:
2.12. Transformations