## Tuesday, July 22, 2014

### #July2014Challenge: Coordinate Algebra and Transformations

I was recently reviewing workshop materials from a transformational geometry session and was directed to the Georgia Deparment of Education website for a unit called Algebraic transformations. This is the fifth unit in their first year (9th grade) course titled Coordinate Algebra.  I found the activity/assessment below on page 109 (answer key on page 103).   The task was to draw some shapes on a coordinate grid, and transform them using 8 different coordinate rules.   Below is a list of similar problems.

What would happen to your shape if you transformed it using the given rule?
1.  (-x,y)
2.  (x,-y)
3.  (x+2,y)
4.  (x, y-3)
5.  (x+1,y-4)
6.  (2x,2y)
7.  (-x,-y)
8.  (3x-2,y+1)

In case you've never seen these rules before, the process is that you take your preimage point (x,y), and substitute the values for x and y into the given coordinate rule to find the image point.  If your set of preimage points creates a shape, then it is easier to see the transformation given by the coordinate rule since your image will form the transformed shape.

1.  Reflect across the y-axis
2.  Reflect across the x-axis
3.  Shift (translate) right two units
4.  Shift (translate) down 3 units
5.  Shift (translate) right 1 unit, down 4 units
6.  Dilate from the origin by a scale factor of 2
7.  Rotate 180 degrees counterclockwise about the origin (or reflect across both axis)
8.  Horizontal stretch and shift, and shift up one unit.  (I am guessing this was an extension of what they learned, so not a precise description).

I am thrilled that the Common Core allows for more time spent on coordinate rules. These rules are somewhat intuitive, and easy to verify using a diagram.  Activities can easily involve multiple representations of knowledge.  This study of coordinate transformations is even included in the 8th grade standards:

8.G.A.3.  Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

My hope is that all of this study of coordinate transformations will help with the understanding of function transformations.  Many of the newer textbooks and resources still teach function transformations via observation.  You might see an introductory activity that asks students to graph each of the following functions and to notice the similarities and differences between each graph.

Students easily make the correct observations, and they can even memorize the rules for vertical translations in this case.  But once we teach students all of the rules for function transformations, it starts to make less sense if the introduction is based on observation instead of reason.  I'd rather review the rules for coordinate transformations each time I introduce a function transformation.  In this case we are transforming a set of points instead of just moving a graph around.  This will also help students see functions as a set of points instead just a graph with a shape.  In my work over the next year I will be looking for textbooks or online resources that utilize this method of teaching function transformations.

I made the following tool to use when practicing coordinate transformations.  To use this tool, enter a coordinate rule for point B that depends on the coordinates of point A.  For example, if Point B= (-x(A), -y(A)), then point B will be rotated 180 degrees about the origin from point A.  You can use the tool below to try this out.  Once you've entered coordinates for point B, drag point A across the semicircle to see the transformation created by the trace of point B.

I'll admit that this tool isn't as polished as I would like it to be. I wanted to share anyway because this is a concept where the use of technology can enhance instruction and practice.  The updated version will be added to the Geogebra Tools list at the top of this blog when available.  For now you can access a download of this tool from the Geogebratube website, and from there you can open with Geogebra and resize the window and turn on the option to view coordinates, etc.