  A Geometric Approach
to Matrix Transformations
using the Parametric Aplet

As any teacher knows, one of the most effective learning strategies is the investigative approach. By adapting the Parametric aplet to work with the Matrix Catalogue, this approach can be used when teaching the use of matrices to perform geometric transformations.

The instructions which follow will demonstrate the process of setting up a copy of the Parametric aplet to display the contents of two point matrices stored in M2 and M3. The matrix M1 will be used for the 2x2 transformation matrix.

The student can then be encouraged to experiment by changing the transformation matrix M1, left multiplying it by M2 to produce a new M3. The aplet can quickly and easily PLOT the resulting shapes in order to view the geometric effect.

Note:  For technical reasons which are not worth going into, this aplet can cause a problem if it is transmitted from one 39G to another using the infrared link.  The problem does not appear in any later models (the 39g+, 39gs & 40gs) as far as I know.  The easiest way to avoid any problems is to delete the four equations (X1, Y1, X2 & Y2 below) before transmitting it and then have the receiver re-enter them. You can also go to my Maths Aplets page and download the programmed version of this aplet.  This has a program attached to it that runs every time the user presses START and re-enters the equations.

Setting up the ApLet

 In the APLET view, RESET the Parametric aplet and then SAVE it under a new name. The example shown right uses the name ‘Transformer’.           Next, change into the Matrix Catalogue in order to set up the matrices.  In this example, an initial transformation matrix in M1 of will be used, together with a point matrix in M2 of which represents with A(1,1), B(2,1) and C(1,3).  The final repeated value of (1,1) is to connect back to the starting point A. ie. To connect: A => B => C => A.   Now change to the HOME view and generate the transformed point matrix M3 by left multiplying M2 by M1. The matrix M3 represents the triangle’s image .     The final stage is to set up the newly created aplet to display the original triangle and its image. Change to the SYMB view and enter the parametric equations shown right. Using these equations, the x and y coordinates will be drawn from rows 1 and 2 respectively of the two point matrices.  In order to make this trick work and to retrieve each of the four columns in turn from the matrices, we have to ensure that only the values 1, 2, 3 & 4 will be used for T. This is done by changing to the PLOT SETUP view and setting TRng and TStep as shown on the right. The x and y axes have also been changed, but this is a matter of choice. Press the PAGE ò button to change to the second page of the PLOT SETUP view. Now üCHK the Connect and Grid settings and unüCHK the Simult setting.       Finally, press PLOT to see your first transformation.    To experiment with other transformations, EDIT the matrix in M1 and repeat the calculation in the HOME view (just press ENTER if it is still the last calculation).   When you press PLOT this time nothing will happen because the aplet does not ‘realise’ that the matrix M3 has changed. Just press PLOT again to force a re-draw showing the new data.  Other shapes than the triangle shown can be used, simply by altering M2. If more than four columns are needed then the value of Tmax will need to be changed in PLOT SETUP. For example, using a quadrilateral would require five points (remember that you must reconnect back to the first point) and Tmax will need to be changed to 5. It is best to avoid using shapes that are too regular since it makes it harder to tell when they have been rotated. For similar reasons it is best to avoid points which are on or straddling an axis unless you want to illustrate stationary points.

Classroom use

The degree of guidance which should be given to a class will obviously depend on their level of ability. The group with which I first used this aplet was very able and, once shown how to set up the Transformer aplet, immediately begin experimenting with different transformation aplets. The only guidance I gave them was to suggest that they confine their investigations initially to placing numbers only on the diagonals. I challenged them to record their matrices on the board as they discovered them and they were able to find all except the shear matrices within 20 minutes.

If it has been a while since the students have encountered geometric transformations, some guidance may be needed on how to explain the effects in geometric terms. For example, while they may be quick to see reflection in the axes, I had to point out to them that reflection in other lines such as y = x and y = -x was also possible. It was also necessary to explain why a dilation appeared to ‘move’ the triangle as well as enlarging it. Rotation by 90 degrees was quickly discovered and this lead easily to the idea of combining transformations in order to produce rotations of 180 and 270 degrees.

Teachers with slightly less able groups may choose to supply greater guidance. One possible method might be to supply two sets of cards. One set would display a matrix on each card, the other set a geometric transformation on each card. A group of students can be supplied with a shuffled set of these cards and students could use the Transformer aplet to match matrices with transformations.