Monday, 2/29/16: Rotation Art Project

We are now in Unit 4, which is on using transformations to establish congruence, using transformations to establish similarity, the Pythagorean theorem, angle relationships in parallel lines and triangles, and the volume of cylinders, cones, and spheres.

We're starting with transformations, of which there are four types:

• translations (slides)
• reflections (flips)
• rotations
• dilations (zooming in or out)

Students, parents / guardians, and tutors, as we move forward, there are going to be two key tools that will help students with transformations (in addition to the normal internet and textbook resources):
First, I've given each student a section of tracing paper (to be kept in the plastic sleeve with the class assignment sheet and used for the entire unit).  For many students this will be of significant help, using the tracing paper to mimic the motion.  Second, look for patterns in the ordered pairs (x and y combinations) pre-image (the before) to image (the after).  From the very beginning, I am going to be making my students record all of the ordered pairs, before-and-after.  The patterns they will discover there will really help them.

I explained tonight's homework in class tonight.  Students are to take the set of pre-image coordinates that they created from their first art project (the reflection across the x-axis) and create a new image set of coordinates, that will represent a 90-degree clockwise notation.  Each new ordered pair in the image set of coordinates will be of the form (y,-x), where x represents the original x-coordinate for the corresponding pre-image pair, and y represents the original y-coordinate.  That is to say, looking at all of the ordered pairs in the pre-image column, all the student has to do is write new ordered pairs, with the original y-coordinates (y-numbers) written first, and the original x-coordinates (x-numbers) written second, but with the opposite sign.

For example, say that the original pre-image ordered pairs had these:

(-3, 5)
(-5, 7)
(8, -3)
(8, 9)
(2, -4)

The corresponding ordered pairs would look like this:

pre-image:                   image:

(x,y)                            (y,-x)

(-3,5)                           (5, 3)
(-5,7)                           (7,5)
(8,-3)                           (-3,-8)
(8,9)                            (9,-8)
(2,-4)                           (-4,-2)

If you look at all of the y's in the first column, you'll see that they're written as is, but first, in the new second column.  If you look at all of the x's in the first column, you'll see that they are written second in the new second column, but with opposite signs for each.

Then students would graph the new ordered pairs (in the image column) that they just created.  This is how computers manipulate the pixels to render a 90-degree rotation clockwise.  Tonight's homework is to create new ordered pairs off of the original ordered pair (pre-image) data set, and then create the new graph which should be a 90-degree rotation to the right.