Investigating generalizations

After investigating connections between rotations and reflections I started wondering about generalizations. I felt this would be a good way for students to practice working with rotations and gain some better insight into the difference between a procedure working in specialized situations or working in general.

I posed the question to the class whether the methods they used in the investigation (connecting line segments created from two different pairs of points between the pre-image and image) would always identify the center of rotation. I also asked if the two reflecting lines always provide a way to determine the angle of rotation and the center of rotation.

Tables investigated and discovered that the process of connecting two line segments as per the investigation did not work in general. This is a valuable lesson because too often students learn some procedure and start applying without regard to when it is appropriate to apply.

The second piece led to the conclusion that the center of rotation is at the point of intersection of the two reflecting lines and can be used to help determine the degree of rotation.

I then posed the reverse of this, given a pre-image and image figure, how do you determine the reflecting lines and from them identify the center of rotation and the degree of rotation.

I intend to use this as a springboard back to constructions. I am hoping that students make some connection to bisectors (perpendicular bisectors in particular).

Also, my intention is to use MS Paint to investigate dilations. I can use the coordinates of the figure and then change the size by percentage and look at the new coordinates. I think this will bring dilations to life and provide a more understandable setting as to what is going on and why.