In teaching geometry, lately I’ve become a fan of compositions of transformations–thinking of a Transformation like a reflection about the x-axis as a function that takes one point and outputs its reflection point. Those rules for reflection are easy, especially about the lines y=0, x=0, and y=x.
about y=0 … (x,y)–>(x,-y)
about x=0 … (x,y)–>(-x,y)
about y=x … (x,y)–>(y,x) The inverse function reflection!
And a few years ago, I realized that a reflection about x=0, then reflected about y=0 gave a 180 degree rotation, with the rule (x,y) –>(-x,-y) because 2 flips equals a rotation. And the angle of rotation will be twice the angle between the two axes of reflection (twice 90 is 180).
But I always struggled to understand the 90-degree rotations until now.
Rotate 90 degrees clockwise (x,y)–>(y,-x)
90 equals 45 times 2. So there’s a 45 degree angle between y=x and y=0. So I put (x,y) through the first rule, and it becomes (y,x). Then I put (y,x) through the second rule for reflecting about y=0, and get (y,-x) because reflecting about y=axis means make the second coordinate the opposite sign.
Not sure how useful this is to teaching regular level geometry classes, but it helps me know it, and it might help more kids in honors level geometry classes.