Using Transformer2D is considerably different from using GridMaster, in which all actions took place on a single graphical interface. In Transformer2D, the interface has been partitioned into different regions, each having its own purpose. The yellow box (upper left) controls the column vectors defining the matrix of transformation -- the green vector is the first column and the blue vector is the second column. By grabbing and moving the ends of these two vectors, you can construct any 2x2 matrix. Clearly, varying the matrices' column vectors changes only the matrix. This encourages students to think about Ax as a linear combination of the columns of A and hopefully begin to recognize that the column space is the range of the transformation.
Below the yellow box is a box that controls the vector x. As you move x (the red vector) about the domain of the transformation, you can watch the image T(x) (the magenta vector) change in the large area to the right of the screen depicting the codomain. Partitioning the sketch in this way allows students to investigate separately the impact of changes to the matrix of transformation and of the value of the x-vector. However, in order not to focus solely on the transformation of a single vector, we have provided a variety of other built-in graphical elements in the domain: a unit circle, a unit square, a unit grid, and an adjustable quadrilateral. (Change the shape of the quadrilateral by moving its vertices.) Students can view the transformations of each of these shapes by selecting the corresponding button in the domain sector. In addition, students can turn on or off a tracing feature that marks out the path of the transformed vector T(x) as x is varied. This feature allows students to carry out freehand investigations.
Open Transformer2D in new window
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Notes:
This sample activity provides a guided exploration of a particular matrix of transformation and includes a set of questions that can be asked for any other matrix of transformation.
Given your exploration, answer the following questions:
In addition, Transformer2D can be used to find a matrix of transformation with particular properties such as:
Next or page: 7. Student Responses, Part 1