Defining congruence using CCSS is not the easiest concept that I have ever taught to a group of students…and I am not convinced that I have it all right. My students have always come in to their high school geometry course thinking of two congruent figures as those with the “same size and shape”. So instead, we are asked to define two objects as congruent when there is a rigid motion that maps one object onto the other. I’ve mulled over how to introduce this to students (in the back of my mind, most of the summer), and I decided to use a document from Geometry Nspired to begin.
Transformations: Translations actually begins with the pre-image and image on top of each other, directing students to move point C to begin the translation. I decided instead to give the students the document with the triangle already translated and ask students how we know the two images are congruent.
Even though we can see that perimeters, areas, and some corresponding side lengths are equal, we decided that the two triangles were congruent because we could grab and move C’ to translate triangle A’B’C’ onto ABC.
Triangles ABC and A’B’C’ are congruent because there is a rigid motion that maps one onto the other.
After exploring what happens to coordinates during a translation and introducing more sophisticated notation than “left 2, up 3” to talk about translations, I asked students to justify that one triangle was a translation of another:
Many students noted that they missed the x-y coordinate plane. How could they show that this was a translation without being able to talk about “x units right and y units down”? Several students began to measure sides, angles, perimeter, and area of the triangle. But just because our previous experience with those measurements tells us the triangles are congruent doesn’t mean one has to be a translation of the other.
Some began to think about the mapping of A to A’, B to B’, and C to C’.
Is it enough to show that segments AA’, BB’, and CC’ are congruent? Is there some other relationship between the lines that contain those segments?
One student noticed that the lines that contain the segments are parallel to each other. How would we show that they are parallel? Students remembered that we could use slopes to show parallel lines…but how else could we show that the lines are parallel? Another student said something about the distance between the lines. So we talked about what “distance between two lines” means.
And so, the journey continues…