a) Verify experimentally the properties of dilations given by a center and a scale factor: a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
b) Verify experimentally the properties of dilations given by a center and a scale factor: the dilation of a line segment is longer or shorter in the ratio given by the scale factor.
INTERPRETATION OF OBJECTIVE - G.SRT.A.1
This objective is all about learning the fundamental properties of dilation. Just like how in G.CO.A.4 we defined the isometric transformations and then looked closely at the characteristics and properties - that is what we are doing here. We want to understand how dilations move points and what properties follow from this motion in the plane.
(1) The student will determine the properties of dilation.
(2) The student will be able to dilate when the center of dilation is in, on and out of the shape.
(3) The student will be able dilate when given a center of dilation and a scale factor.
(4) The student will be able to determine the center of dilation and the scale factor from a diagram.
(5) The student will be able dilate using both positive and negative scale factors.
(6) The student will be able to construct a dilation.
(7) The student will be able to use the dilation coordinate rules for dilations using any center of dilation.
THE BIG IDEA
The big idea is similarity!!! The isometric transformations all lead to the concept of congruence and the non-isometric transformation, the dilation, leads us directly to similarity. After learning these properties we will see how parallel lines are formed in dilations and how lengths are all scaled proportionally - thus similar. Gaining the foundational ideas here will pay off greatly later!!
TRAPS & PITFALLS
In general I think dilation is not very difficult but a few circumstances seem to always catch students. One such situation is dilating a line when the center of dilation is on that line. This is very difficult for students to visualize - Unless they have a strong conceptual understanding from the definition this will be very abstract.
Determining the coordinates of an image when you are dilating from a center of dilation that is not the origin can also be quite challenging for students. They find that the very basic multiplication rule, no longer works and their slope technique is weak. Actually a little extra time spent on the general rule for all centers of dilation will unlock a later objective, G.GPE.B.6, when we are asked to partition a line seGment into a given ratio.
In G.CO.A.2 we discussed what an isometry is. At that time we discussed that dilation WAS NOT isometric because it changed the size of the shape. We knew that at some stage we would look at the properties of dilation up close.
Students will use dilations in lots of places, Similarity, Parallel Line relationships, geometric mean, special right triangles, trigonometry and partitioning a line segment to name a few. These early concepts and properties found in dilating lay the basis for lots of ratio related mathematics later in the course.
MY REFLECTIONS (over line l)
After having taught this I realized that student struggled with some of the conceptual questions concerning dilating a line, when the center of dilation is either on the line or off the line. I realized that almost everything I dilated was a triangle, and the center of dilation was usually outside that triangle. They got that but when I put a point on a line and asked them to dilate it by a scale factor of 2, I saw everything under the sun appear. I was so surprised.... so this might be a long way of saying - "Hit the properties and definition of dilation hard!!!" Time spent here will pay off later, when working with ratios and similar triangles.