What do we need for a reflection?
An object. And a line.
My students have some experience with reflections before they get to my high school geometry course…so it didn’t take them long to articulate what we need for a reflection or to predict where the image of ΔABC would be when it is reflected about line XY.
Pretty quickly, I let them use their dynamic technology to reflect ΔABC about line XY.
And I asked them to write down everything they could find that was congruent. This didn’t take as long as it did for the translation, because they got that a reflection is a rigid motion, and so ΔABC is congruent to ΔA’B’C’. And then that means that segment AB is congruent to segment A’B’. And the perimeter of ΔABC is equal to the perimeter of ΔA’B’C’. And the areas of the triangles are equal. And the corresponding angles are congruent. But what else has to be congruent? Someone suggested that the distance from C to line XY will be the same as the distance from C’ to line XY. “Yes!” (I thought to myself as I hid my excitement.) But as we stopped to have a conversation about what one means when they talk about the distance from C to line XY, I realized that my students had no idea what it means mathematically to talk about the distance from a point to a line. I asked if it matters where we draw the distance from C to line XY. All 30 students shook their head no.
So can any of the red segments drawn from C to line XY represent the distance from C to the line? All 30 students shook their head yes.
Are all of those segments the same length? All 30 students shook their head no. And finally someone talked about “seeing” a right triangle (look for and make use of structure) – and said that segment CF is longer than segment CE because it is the hypotenuse of a right triangle. At some point we talked about the phrase “as the crow flies” – and the fastest way to get from one location to another, which the students said is a straight line (to which I reply every single time that all lines are straight in our geometry).
It was time to define the distance from a point to a line as the length of the segment that lies on the line that is perpendicular from the point to the line. Other relationships became evident as well. E is the midpoint of segment CC’. Line XY is the perpendicular bisector of segment CC’. Angle CED is congruent to angle CEF, and they are both right angles. And similarly if we talk about segment AA’ or segment BB’ instead.
We all learned something in class today. I know more about student misconceptions for the distance from a point to a line. My students know more about the distance from a point to a line than they did before class. And I hope that my students know more about the distance from a point to a line because of the context and conversation we had than if I had started class with notes and examples about the distance from a point to a line & asked them to work a few problems.
And so the journey continues ….
Note: We used the Transformations – Reflections lesson from Geometry Nspired as a guide for part of our exploration in this lesson. http://education.ti.com/en/timathnspired/us/detail?id=A370588B92164483B5C08C3E3833A443&t=E51B8F52AF8A4099A8204BE3B9ED5BD6