I only managed to get through one unit this week which I was a bit disappointed about. I didn’t expect to get through two, but I thought I’d at least be able to start a second unit which didn’t happen. It turned out the unit I did manage to get through, Transformations, took more time than I thought it would BUT I’m happy to say I just passed the unit test with a 100% score.

Based on what I knew about the word transformation, if you had asked me what a transformation was in terms of geometry I would have guessed it meant changing a shape. I would have essentially been correct, however there are nuances to geometrical transformations that I wouldn’t have been able to tell you about. The following is a list of the types of transformations I learned about and how they’re applied:

• Translation
  • Moving a all the points of a line/shape on a graph the same direction and distance simultaneously.
  • It’s essentially just moving the shape around the graph without altering the shapes’ dimensions.
• Rotation
  • Rotating a shape on a graph around one point. The point of rotation can be outside the shape, inside the shape, or on the perimeter or a vertex of the shape.
  • Rotating a shape counter-clockwise is written in positive degrees (ex. 45 degrees, 140 degrees, etc.) and rotating a shape clockwise is written in negative degrees (ex. -25 degrees, -246 degrees).
• Reflection
  • Flipping a point, line, or shape across a “line of reflection”.
  • Could be thought of as a mirror image of the source shape.

All of the above transformations are known as “rigid” transformations which means the lengths and angles of each shape are preserved after the transformation takes place.

Two other important terms to know when describing transformations are the words “source” and “image”. The source shape refers to the original shape and its’ original coordinates. The image shape is the shape and its’ new coordinates after the transformation has taken place.

The only non-rigid transformation that I learned about is called dilation.

• Dilation
  • Scaling a shape up or down (i.e. making a shape bigger or smaller) via a point of dilation.
  • Angles and the ratio of lines are preserved.
  • Similar to a rotation, the point of dilation can be inside the shape, on the perimeter or vertex, or outside the shape. The shape “stretches” or “shrinks” in relation to that specific point.
  • The “scale factor” of a dilation refers to how much the shape is stretched of shrunk. If the scale factor is 3, the coordinates of the source are pushed away from the point of dilation by a factor of 3 to get the coordinates of the image. If the scale factor is 1/3, the coordinates of the source are brought closer to the point of dilation by a factor of 3 to get the coordinates of the image. Bringing the vertex points closer together or further apart by a factor of 3 also causes the lines of the shape grow or shrink by a factor of 3. It does not change the angles, however.

Annoyingly, in some of the exercises I was given questions on “stretching” a shape which is also a non-rigid transformation. I was annoyed because I wasn’t shown any videos or given any tutorials on stretching shapes and, though I managed to get the gist of it to pass the exercises, I don’t have a thorough understanding of how stretching a shape works.

When drawing transformations, the common way to label the vertexes of the image is by using an apostrophe (ex. the source vertex A turns in the image vertex A’). Writing the apostrophe next to the letter refers to it as “A prime” or “B prime”, etc.

When rotating shapes, I was shown formulas to quickly figure out the coordinates for a shapes image when rotating the shape 90 degrees, -90 degrees, and 180 degrees. The formulas only work, however, when the rotation is done around (0, 0) which is also known as “the origin”. They are:

• 90 degrees – (x, y) turns to (-y, x)
  • Ex. (3, 4) = (-4, 3)
• -90 degrees – (x, y) turns to (y, -x)
  • Ex. (-4, 5) = (5, 4)
• 180 degrees – (x, y) turns to (-x, -y)
  • Ex. (14, -13) = (-14, 13)

I also learned a formula to determine the line of reflection when given the (x, y) coordinates of a source and image shape which is ((x + x’)/2, (y + y’)/2).

It just hit me that I’m starting Week 20 this coming week. Although it has taken me much longer than I first anticipated, I’m happy with the progress I’ve made and am proud that I’ve been able to stick to it. The hardest part of this whole thing has been sticking with it week after week. This coming week I’ll be taking on the unit Congruence (0/400 M.P.). Although it’s only 400 M.P., there are a fair amount of videos/exercises. That said, my hope is to get through this unit by midweek and hopefully the following unit, Similarity (400/700 M.P.), by the end of the week.